Principal curvature

In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point.

At each point p of a differentiable surface in 3-dimensional Euclidean space one may choose a unit normal vector.

Here the curvature of a curve is by definition the reciprocal of the radius of the osculating circle.

The curvature is taken to be positive if the curve turns in the same direction as the surface's chosen normal, and otherwise negative.

For hypersurfaces in higher-dimensional Euclidean spaces, the principal curvatures may be defined in a directly analogous fashion.

Similarly, if M is a hypersurface in a Riemannian manifold N, then the principal curvatures are the eigenvalues of its second-fundamental form.

The lines of curvature or curvature lines are curves which are always tangent to a principal direction (they are integral curves for the principal direction fields).

In the vicinity of an umbilic the lines of curvature typically form one of three configurations star, lemon and monstar (derived from lemon-star).

[2] These points are also called Darbouxian Umbilics (D1, D2, D3) in honor of Gaston Darboux, the first to make a systematic study in Vol.

The implication of such an orientation frame at each surface point means any rotation of the surfaces over time can be determined simply by considering the change in the corresponding orientation frames.

This has resulted in single surface point motion estimation and segmentation algorithms in computer vision.